The Women’s Equality party has promised to fight for free childcare from when a baby is nine months old, fully equal parental leave, baby-changing areas available to men and women, and the criminalisation of those who pay prostitutes for sex – among a series of other proposals.

The party, led by former Reuters journalist Sophie Walker, set out its policy positions at a launch in London. It seeks to achieve equality in politics, business, education, pay, parenting and the media, as well as an end to violence against women.

Despite having only been formed in March by comedian Sandi Toksvig and journalist Catherine Mayer, it now has 45,000 members and supporters across 65 branches, compared with Ukip’s most recent claim to have “more than 40,000” members and the Green party’s 65,000.

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Last week, Walker revealed in the Guardian that one of the party’s flagship policies would be calling for a gender quota system to select MPs at the next two elections to achieve equal representation in the House of Commons by 2025.

The policy document calls for:

Listed companies to ensure at least half of their board members are women by 2025.

Equal parental leave of six weeks for each parent on 90% of pay.

The married tax allowance to be scrapped, with the savings diverted into funding for rape crisis centres.

More legal protections for cohabiting couples with children who have been together for more than two years.

Baby-changing facilities open to both genders.

Government-funded childcare for all children at nine months old, with the first 15 hours free and all extra time available at £1 per hour.

The Nordic model for prostitution where those who pay for sex are criminalised, while sex workers are never prosecuted.

Compulsory age-appropriate sex education, including about consent, in all state funded schools.

Gender audits of the curriculum to ensure children are taught about women’s achievements.

Launching the policies, Walker said: “We want women to realise their potential, and we want to do it by providing a system of childcare that doesn’t cost the earth – and doesn’t have to be waited for.

“We believe that government-funded childcare should be available for all children from the end of paid parental leave at nine months, and we believe this is a key area for government investment.

“We would fully fund this by introducing a single rate of tax relief on pension savings, which would free up around £6.5bn pounds.

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“This supports women as they build their careers and start families, but it also protects women who are low earners and part-time workers to save for their pensions.”

The party also challenged the prime minister, David Cameron, and Labour leader, Jeremy Corbyn, to match their words on gender equality with action, as it highlighted the £245bn pay gap between men and women.

It said men’s earnings in the UK totalled £516bn a year, compared with women’s pay of £271bn – due to women being paid less than their male counterparts, being forced out of work by expensive childcare and having to stay at home to look after children and elderly parents.

Cameron told the Conservative party’s autumn conference that he wanted to end all discrimination saying: “You can’t have true opportunity without real equality.”

Walker said: “These are good comments. But they’re just comments. They’ve been talking about delivering equality for far too long. It seems to me that they are more interested in claiming the right to deliver equality than actually deliver it.

“Where’s the action? Their only action is to shoot down any other party’s talk about how they might deliver equality – reducing our right to participate to a cynical messaging contest.”